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# Non-invertible anomalies and Topological orders

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It has been realized that anomalies can be classified by topological phases in one higher dimension. Previous studies focus on ’t Hooft anomalies of a theory with a global symmetry that correspond to invertible topological orders and/or symmetry protected topological orders in one higher dimension. In this talk, I will introduce an anomaly that appears on the boundaries of (non-invertible) topological order with anyonic excitations [1]. The anomalous boundary theory is no longer invariant under a re-parametrization of the same spacetime manifold. The anomaly is matched by simple universal topological data in the bulk, essentially the statistics of anyons. The study of non-invertible anomalies opens a systematic way to determine all gapped and gapless boundaries of topological orders, by solving simple eigenvector problems. As an example, we find all conformal field theories (CFT) of so-called minimal models’’, except four cases, can be the critical boundary theories of Z_2 topological order (toric code). The matching of non-invertible anomaly have wide applications. For example, we show that the gapless boundary of double-semion topological order must have central charge c_L=c_R >= 25/28. And the gapless boundary of the non-Abelian topological order described by S_3 topological quantum field theory can be three-state Potts CFT, su(2)_4 CFT, etc. [1] WJ, Xiao-Gang Wen, arXiv: 1905.13279, Phys. Rev. Research 1,033054